#195535

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6,8,11)(3,9,10,12)(17,18)(19,27,31,41)(20,32,34,24)(21,26)(22,25)(23,30,39,35)(28,37)(29,43)(33,42,36,40)(38,44)(45,48)(46,47);(1,2,4,11,5,8,13,6)(3,7,12,16,10,15,9,14)(17,19,28,30,22,31,21,35)(18,23,37,27,25,39,26,41)(20,29,33,46,34,45,36,44)(24,38,40,43,32,47,42,48);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,13)(11,16)(17,20,22,34)(18,24,25,32)(19,29,31,45)(21,36,28,33)(23,38,39,47)(26,42,37,40)(27,43,41,48)(30,46,35,44);(2,8)(6,11)(7,15)(14,16)(18,25)(23,39)(24,32)(26,37)(27,41)(38,47)(40,42)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102731);